EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6277
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Nonlinear Advanced Differential Equations: Improving
Some Properties of Positive Solutions and Their

Applications

Wedad Alhaysuni1, Saleh Fahad Aljurbua1,∗, Osama Moaaz1

1 Department of Mathematics, College of Science, Qassim University, P. O. Box 6644,
Buraydah, 51452 Saudi Arabia

Abstract. This study examines the oscillatory performance of solutions of functional differential
equations with an advanced argument. Equations of the advanced type have not received as much
study as equations with delay. We deduce some new monotonic properties of the positive solutions
of the studied equation. Then, we use these properties to obtain criteria that test the oscillatory
nature of the solutions. We apply the results to some special cases, compare them with previous
results, and analyze the results to demonstrate the novelty and importance.

2020 Mathematics Subject Classifications: 34C10, 34K11

Key Words and Phrases: Differential equations, advanced argument, oscillatory performance,
noncanonical case

1. Introduction

Advanced differential equations (ADEs) are necessary for defining a wide range of
systems that rely on current values and future predictions. These equations are used in a
variety of phenomena, including population dynamics, economic models, and mechanical
control systems; see [1–3]. Obtaining a closed solution to these equations is difficult, so
we use oscillation theory, among the most important subfields of qualitative theory; see
[4, 5].

The primary goal of this study is to examine how second-order ADEs exhibit oscillatory
behavior. Here, we consider the ADE(

ρ (s)
[
x′ (s)

]κ)′
+ q (s)xκ (h (s)) = 0, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6277

Email addresses: 441212393@qu.edu.sa (W. Alhaysuni),
s.aljurbua@qu.edu.sa (S. F. Aljurbua), o.refaei@qu.edu.sa, o moaaz@mans.edu.eg (O. Moaaz)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 2 of 11

where s ≥ s0, κ ≥ 1 is a ratio of two integers, ρ, q ∈ C ([s0,∞) , [0,∞)) , ρ (s) > 0,
h ∈ C ([s0,∞) ,R) , h (s) ≥ s, h′ (s) ≥ 0, and∫ ∞

s0

ρ−1/κ (ϱ) dϱ <∞.

For convenience, we define the integral operator

Λ [f (·) ; u, v] :=
∫ v

u
f (ϱ) dϱ,

for v ≥ u ≥ s0 and f ∈ C ([s0,∞)), and the function

P (s) := Λ
[
ρ1/κ; s,∞

]
.

A function x ∈ C1 ([sx,∞)), sx ≥ s0, is called a proper solution of (1) if it has the features
sup {|x (s)| : s ≥ s1} > 0 for s1 ≥ sx, ρ [x

′]κ ∈ C1 ([sx,∞)), and satisfies (1) for all sx ≥ s0.
In this study, x is said to be an oscillatory solution if it has a sequence of zeros {sn}∞n=0

such that limn→∞ sn = ∞. An equation is said to be oscillatory if all of its solutions
exhibit oscillatory behavior.

While the oscillation of equation (1) in the context of delay, i.e., h (s) ≤ s, has been
extensively studied (see [6–10]), research on the oscillation of ADEs, where h (s) ≥ s,
remains relatively limited. However, interest in this area has grown significantly in recent
decades due to its increasing relevance; see [11–15]. For related results in fractional models,
see [16, 17] and references therein.

In 1981, Kusano and [18] demonstrated the oscillation of the equation(
ρ (s)x′ (s)

)′
+ q (s)x (h (s)) = 0 (2)

can be inferred from the oscillation of the corresponding ODE(
ρ (s)x′ (s)

)′
+ q (s)x (s) = 0. (3)

Equation (3) is oscillatory if

ρ(s)Λ [q; s,∞] ≥ δ0 >
1

4
. (4)

where δ0 is a constant; however, this condition is better suited for ODEs since it loses
information on the value of advanced argument h (s).

In [19], Džurina established oscillation criteria for equation (2), based on the related
ODE (3), and he attempted to modify and improve the condition 4, which fails if (δ0 ≤ 1

4).
He used the condition

ρ(s)Λ [q; s,∞] ≥ δ0 > 0, (5)

which ensures that the equation

(ρ(s)x′(s))′ +

(
ρ(h(s))

ρ(s)

)δ

q(s)x(s) = 0 (6)



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 3 of 11

is oscillate.
In 2020, Bohner et al. [12] developed the previous works and investigated the oscillation

of equation (2) in the noncanonical case. They transformed it into an equation in canonical
form.

2. Main Results

As usual in oscillation theory, we focus on studying the properties of the positive
solutions of equation (1). We directly exclude the negative solutions due to their symmetry
with the positive solutions. We indicate that the solution belongs to the class of eventually
positive solutions of (1) by the expression x ∈ X+.

2.1. Auxiliary Lemmas

We begin the main results with the following lemmas, which deduce asymptotic and
monotonic properties for x ∈ X+ and transform the equation (1) into a linear equivalent
form (in oscillatory features).

Beginning by the following lemma which provides a criterion that rules out the exis-
tence of positive increasing solutions under certain conditions. This insight plays a crucial
role in the subsequent analysis of oscillatory behavior.

Lemma 1. Assume that x ∈ X+. Then x is decreasing and converging to zero if

Λ

[
1

ρ1/κ
(Λ [q; s0, s])

1/κ ; s0,∞
]
= ∞. (7)

Proof. Assume that x ∈ X+. Therefore, there is a s1 ≥ s0 such that x (h (s)) > 0, and
so (

ρ (s)
[
x′ (s)

]κ)′
= −q (s)xκ (h (s)) ≤ 0. (8)

Hence, x′ is of fixed sign.
Now, let x′ is positive for s ≥ s1. Then, x → x0 as s → ∞, where x0 is a positive

constant. We also conclude that (x ◦ h) ≥ x0, for s ≥ s1. Before proceeding to prove that
x is decreasing, we note that (7), with the fact that P ′ (s) ≤ 0, guarantees this

Λ [q; s0,∞] = ∞. (9)

Applying Λ [(·) ; s1,∞] on (8), we obtain

ρ (s1)
[
x′ (s1)

]κ ≥ Λ [q · (xκ ◦ h) ; s1,∞]

≥ xκ0Λ [q; s1,∞] .

which contradicts to (9). So, x′ is negative for s ≥ s1.
The positivity and decreasing of x ensures its convergence to x1 ≥ 0. Suppose that

x1 > 0. Applying Λ [(·) ; s1, s] on (8), we find

ρ (s)
[
x′ (s)

]κ ≤ −Λ [q · (xκ ◦ h) ; s1, s]



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 4 of 11

≤ −xκ1Λ [q; s1, s] ,

or

x′ (s) ≤ −x1
1

ρ1/κ (s)
(Λ [q; s1, s])

1/κ . (10)

Applying Λ [(·) ; s1,∞] on (10), we arrive at

x (s1) ≥ x1Λ

[
1

ρ1/κ
(Λ [q; s1, s])

1/κ ; s1,∞
]
,

which contradicts to (7). Therefore, x1 = 0.
The proof is complete.

Lemma 2. Assume that (7) holds. Then

d

ds

( x
P

)
≥ 0 (11)

and (
ρ1/κ · x′

)′
+

1

κ
q · (P ◦ h)κ−1 · (x ◦ h) ≤ 0. (12)

Proof. Assume that x ∈ X+. We have

−x (s) = Λ
[
x′; s,∞

]
= Λ

[
ρ1/κ · x′

ρ1/κ
; s,∞

]
≤ ρ1/κ (s)x′ (s)P (s) .

So,
d

ds

( x
P

)
=

P · ρ1/κ · x′ + x

ρ1/κ · P2
≥ 0.

Thus, from the fact that h (s) ≥ s, we have

(x ◦ h)
(P ◦ h)

≥ x

P
≥ ρ1/κ ·

(
−x′

)
, (13)

and so (
(x ◦ h)
(P ◦ h)

)1−κ

≤
[
ρ ·
(
−x′

)κ]1/κ−1
. (14)

Now, it follows from (1) and (14) that(
ρ1/κ ·

(
−x′

))′
=

([
ρ ·
(
−x′

)κ]1/κ)′
=

1

κ

[
ρ ·
(
−x′

)κ]1/κ−1 (
ρ ·
(
−x′

)κ)′
≥ 1

κ

(
(x ◦ h)
(P ◦ h)

)1−κ

q · (xκ ◦ h)

=
1

κ
q · (P ◦ h)κ−1 · (x ◦ h) .

The proof is complete.



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 5 of 11

Lemma 3. Assume that (7) holds. Then(
ρ1/κ · P2 ·

( x
P

)′)′
+

1

κ
q · P · (P ◦ h)κ−1 · (x ◦ h) ≤ 0. (15)

Proof. Assume that x ∈ X+. It follows that

d

ds

(
ρ1/κ · P2 · d

ds

( x
P

))
=

d

ds

(
ρ1/κ · P2 ·

(
P · x′ + ρ−1/κx

P2

))

=
d

ds

(
P · ρ1/κ · x′ + x

)
= P ·

(
ρ1/κ · x′

)′
− ρ−1/κ · ρ1/κ · x′ + x′

= P ·
(
ρ1/κ · x′

)′
.

So, equation (1) reduces to(
ρ1/κ · P2 ·

( x
P

)′)′
≤ −1

κ
q · P · (P ◦ h)κ−1 · (x ◦ h) .

The proof is complete.

2.2. Oscillatory performance of solutions

The following theorem tests the oscillatory performance of solutions of equation (1)
using the comparison technique with first-order equations.

Theorem 1. Assume that (7) holds. Then, equation (1) is oscillatory if

lim inf
s→∞

Λ

[
Λ [q · P · (P ◦ h)κ ; s,∞]

ρ1/κ · P2
; s, h (s)

]
>
κ

e
. (16)

Proof. Assume the contrary that x ∈ X+. From Lemma 2 and 3, we obtain that (11)
and (15) hold for s ≥ s1 ≥ s0. Applying Λ [(·) ; s,∞] on (15), we get

ρ1/κ (s)P2 (s)

(
x (s)

P (s)

)′
≥ 1

κ
Λ

[
q · P · (P ◦ h)κ · (x ◦ h)

(P ◦ h)
; s,∞

]
,

which with (11) gives

ρ1/κ (s)P2 (s)

(
x (s)

P (s)

)′
≥ 1

κ

x (h (s))

P (h (s))
Λ [q · P · (P ◦ h)κ ; s,∞] ,

Setting w := x/P, we have that w is a positive solution of

w′ (s) ≥ 1

κ

Λ [q · P · (P ◦ h)κ ; s,∞]

ρ1/κ (s)P2 (s)
w (h (s)) .



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 6 of 11

Using Lemma 2.3 in [20], the equation

w′ (s)− 1

κ

Λ [q · P · (P ◦ h)κ ; s,∞]

ρ1/κ (s)P2 (s)
w (h (s)) = 0

has also a positive solution, which contradicts to (16); see Theorem 1 in [21].
The proof is complete.

Based on Riccati’s approach, we test the oscillatory behavior of equation (1) in the
following theorem:

Theorem 2. Assume that (7) holds. Then, equation (1) is oscillatory if there is a ψ ∈
C ([s0,∞) , (0,∞)) such that

lim sup
s→∞

P (s)

ψ (s)
Λ

[
1

κ
ψ · q · (P ◦ h)κ

P
− 1

4

ρ1/κ · (ψ′)2

ψ
; s1, s

]
> 1. (17)

Proof. Assume the contrary that x ∈ X+. From Lemma 2, we obtain that (11) and
(12) hold for s ≥ s1 ≥ s0.

Now, we define

H := ψ ·

(
ρ1/κ · x′

x
+

1

P

)
> 0; see (13). (18)

Then,

H ′ =
ψ′

ψ
·H + ψ ·

((
ρ1/κ · x′

)′
x

− ρ1/κ · (x
′)2

x2
+

1

ρ1/κ · P2

)
,

which with (12) and (18) yields

H ′ ≤ ψ′

ψ
·H − 1

κ
ψ · q · (P ◦ h)κ−1 · (x ◦ h)

x

− 1

ψ · ρ1/κ

(
H − ψ

P

)2

+
ψ

ρ1/κ · P2
.

From (11), we obtain

H ′ ≤ −1

κ
ψ · q · (P ◦ h)κ

P
+
ψ′

ψ
·H − 1

ψ · ρ1/κ

(
H − ψ

P

)2

+
ψ

ρ1/κ · P2
. (19)

We define

F :=
ψ′

ψ
·H − 1

ψ · ρ1/κ

(
H − ψ

P

)2

.

So, F gattains its maximum value at

ψ

P
+

1

2
ψ′ρ1/κ,



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 7 of 11

and

F ≤ ψ′

P
+

1

4
ρ1/κ · (ψ

′)2

ψ
.

Therefore, (19) becomes

H ′ ≤ −1

κ
ψ · q · (P ◦ h)κ

P
+

1

4
ρ1/κ · (ψ

′)2

ψ
+
ψ′

P
+

ψ

ρ1/κ · P2

= −1

κ
ψ · q · (P ◦ h)κ

P
+

1

4
ρ1/κ · (ψ

′)2

ψ
+

(
ψ

P

)′
. (20)

Applying Λ [(·) ; s1, s] on (20), we get

Λ

[
1

κ
ψ · q · (P ◦ h)κ

P
− 1

4

ρ1/κ · (ψ′)2

ψ
; s1, s

]

≤
(
H (s1)−

ψ (s1)

P (s1)

)
−
(
H (s)− ψ (s)

P (s)

)
.

This implies that

Λ

[
1

κ
ψ · q · (P ◦ h)κ

P
− 1

4

ρ1/κ · (ψ′)2

ψ
; s1, s

]
≤ −ψ (s)

ρ1/κ (s)x′ (s)

x (s)
≤ ψ (s)

P (s)
,

or
P (s)

ψ (s)
Λ

[
1

κ
ψ · q · (P ◦ h)κ

P
− 1

4

ρ1/κ · (ψ′)2

ψ
; s1, s

]
≤ 1. (21)

Taking the lim sup of (21), we get at contradiction with (17).
This completes the proof.

3. Discussion and Applications

This section tests the oscillatory performance for some special cases of the studied
equation and compares our results with those previously reported in the literature.

Example 1. Consider the ADE(
s2x′ (s)

)′
+ q0x (h0s) = 0, (22)

where s > 0, h0 ≥ 1 and q0 > 0. Then P (s) = 1/s, and so

Λ

[
1

s2
Λ [q0; s0, s] ; s0,∞

]
= q0Λ

[
s− s0
s2

; s0,∞
]
= ∞,



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 8 of 11

Table 1: Comparison of the oscillation criteria of (22) for special cases of parameters q0 and h0

Our results Previous results
Criterion (23) (24) (25) (26)

(a) q0 = 3 h0 > 26.824 h0 > 0.0833 h0 > 19.078 h0 > 31.005

(b) h0 = 2 q0 > 1.0615 q0 > 0.1250 q0 > 0.4307 q0 > 0.6934

which means that condition (7) is fulfilled.
Now, condition (16) becomes

lim inf
s→∞

Λ

[
Λ

[
q0
1

s

1

h0s
; s,∞

]
; s, h0s

]
=

q0
h0

lim inf
s→∞

Λ

[
1

s
; s, h0s

]
=

q0
h0

ln (h0)

>
1

e
.

From Theorem 1, equation (22) is oscillatory if

q0 >
h0

e ln (h0)
. (23)

On the other hand, by choosing ψ (s) = 1/s, condition (17) reduces to

lim sup
s→∞

Λ

[
q0h0

1

s
− 1

4

1

s
; s1, s

]
=

(
q0h0 −

1

4

)
lim sup
s→∞

Λ

[
1

s
; s1, s

]
> 1.

From Theorem 2, equation (22) is oscillatory if

q0 >
1

4h0
. (24)

Remark 1. According to Theorem 10 in [11] and Theorem 3.4 in [12], equation (22)
oscillates when

q0h
q0
h0

−1

0 >
1

4
. (25)

The results in [13] also confirm the oscillation of equation (22) under the condition

q0
h0

ln (h0) >
1

e

(
1− q0

h0

)
. (26)

The following table, Table 1, illustrates the comparison mentioned in Remark 1 for different
values of q0 and h0. It is clear from this comparison that our results provide more efficient
criteria for testing the oscillatory performance of solutions.

Example 2. Consider the ADE(
esx′ (s)

)′
+ q0e

sx (s+ h0) = 0 (27)



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 9 of 11

where s > 0, h0 > 0 and q0 > 0. Then P (s) = e−s, and so

Λ

[
1

es
Λ [q0e

s; s0, s] ; s0,∞
]
= q0Λ

[
es − es0

es
; s0,∞

]
= ∞,

which means that condition (7) is fulfilled.
Now, condition (16) becomes

lim inf
s→∞

Λ

[
1

e−s
Λ
[
q0e

−s−h0 ; s,∞
]
; s, s+ h0

]
= e−h0q0lim inf

s→∞
Λ [1; s, s+ h0]

= h0e
−h0q0

>
1

e
.

It follows from Theorem 1 that equation (27) is oscillatory if

q0 >
eh0

eh0
. (28)

On the other hand, by choosing ψ (s) = e−s, condition (17) reduces to

lim sup
s→∞

Λ

[
q0e

−h0 − 1

4
; s1, s

]
> 1

From Theorem 2, equation (27) is oscillatory if

q0 >
eh0

4
. (29)

Figure 1 shows the minimum values of q0 at which oscillation occurs according to criteria
(28) and (29). We notice that criterion (28) is more efficient during h0 ≥ 4

e , while
criterion (29) is superior for h0 ∈

[
1, 4e
]
.

Acknowledgements

The authors gratefully acknowledge Qassim University, represented by the Deanship of
Graduate Studies and Scientific Research, on the financial support for this research under
the number (QU-J-PG-2-2025-55703) during the academic year 1446AH / 2024 AD.

Conflict of interest

The authors declare there is no conflicts of interest.



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 10 of 11

Figure 1: Comparison of minimum values of q0 for criteria (28) and (29)

References

[1] Martin Braun and Martin Golubitsky. Differential equations and their applications,
volume 2. Springer, 1983.

[2] Sim Borisovich Norkin et al. Introduction to the theory and application of differential
equations with deviating arguments, volume 105. Academic Press, 1973.

[3] Fathalla A. Rihan. Delay Differential equations and Applications to Biology. January
2021.

[4] Irving R. Epstein and John A. Pojman. An introduction to nonlinear chemical dy-
namics. November 1998.

[5] John Vandermeer. Oscillating populations and biodiversity maintenance. BioScience,
56(12):967, January 2006.

[6] Ravi P Agarwal, Said R Grace, and Donal O’Regan. Oscillation theory for second or-
der linear, half-linear, superlinear and sublinear dynamic equations. Springer Science
& Business Media, 2013.

[7] George E. Chatzarakis and Irena Jadlovsk. Improved oscillation results for second-
order half-linear delay differential equations. Hacettepe Journal of Mathematics and
Statistics, 47(6), November 2017.

[8] Osama Moaaz and Higinio Ramos. On the oscillation of second-order functional
differential equations with a delayed damping term. Applied Mathematics Letters,
page 109464, January 2025.

[9] Jozef Džurina and Irena Jadlovska. A sharp oscillation result for second-order half-
linear noncanonical delay differential equations. Electronic Journal of Qualitative
Theory of Differential Equations, (46):1–14, January 2020.

[10] Taher S Hassan, Clemente Cesarano, Mouataz Billah Mesmouli, Hasan Nihal Zaidi,
and Ismoil Odinaev. Iterative hille-type oscillation criteria of half-linear advanced
dynamic equations of second order. Mathematical Methods in the Applied Sciences,
47(7):5651–5663, 2024.

[11] George E Chatzarakis, J Džurina, and Irena Jadlovská. New oscillation criteria for



W. Alhaysuni, S. F. Aljurbua, O. Moaaz / Eur. J. Pure Appl. Math, 18 (3) (2025), 6277 11 of 11

second-order half-linear advanced differential equations. Applied mathematics and
computation, 347:404–416, 2019.

[12] Martin Bohner, Kumar S Vıdhyaa, and Ethiraju Thandapani. Oscillation of non-
canonical second-order advanced differential equations via canonical transform. Con-
structive Mathematical Analysis, 5(1):7–13, 2022.

[13] Blanka Bacuĺıková. Oscillatory behavior of the second order noncanonical differen-
tial equations. Electronic Journal of Qualitative Theory of Differential Equations,
2019(89):1–11, 2019.

[14] John R Graef. Oscillation of higher order functional differential equations with an
advanced argument. Applied Mathematics Letters, 111:106685, 2021.

[15] Osama Moaaz, Asma Al-Jaser, and Amira Essam. New comparison theorems for
testing the oscillation of solutions of fourth-order differential equations with a variable
argument. Electronic Research Archive, 33(7):4075–4090, 2025.

[16] Muhammad Bilal, Javed Iqbal, Rashid Ali, Fuad A. Awwad, and Emad a. A. Ismail.
Exploring families of solitary wave solutions for the fractional coupled higgs system
using modified extended direct algebraic method. Fractal and Fractional, 7(9):653,
2023.

[17] Muhammad Bilal, Javed Iqbal, Kamal Shah, Bahaaeldin Abdalla, Thabet Ab-
deljawad, and Ikram Ullah. Analytical solutions of the space–time fractional
kundu–eckhaus equation by using modified extended direct algebraic method. Partial
Differential Equations in Applied Mathematics, 11:100832, 2024.

[18] Takasi Kusano and Manabu Naito. Comparison theorems for functional differential
equations with deviating arguments. Journal of the Mathematical Society of Japan,
33(3):509–532, 1981.

[19] Jozef Dzurina. Oscillation of the second order advanced differential equations. Elec-
tronic Journal of Qualitative Theory of Differential Equations, 2018(20):1–9, 2018.

[20] B Bacuĺıková. Properties of third-order nonlinear functional differential equations
with mixed arguments. In Abstract and Applied Analysis, volume 2011, page 857860.
Wiley Online Library, 2011.

[21] Yuichi Kitamura and Takaŝi Kusano. Oscillation of first-order nonlinear differen-
tial equations with deviating arguments. Proceedings of the American Mathematical
Society, 78(1):64–68, 1980.